a2
Department of Biology, Earth and Environmental Science, University
College Cork, Ireland

a3 MACSI, Department of Mathematics and Statistics,
University of Limerick, Ireland

a4 Department of Applied Mathematics, University
College Cork, Ireland

Abstract

Modification of behaviour in response to changes in the environment or ambient
conditions, based on memory, is typical of the human and, possibly, many animal
species.One obvious example of such adaptivity is, for instance, switching to a safer
behaviour when in danger, from either a predator or an infectious disease. In human
society such switching to safe behaviour is particularly apparent during epidemics.
Mathematically, such changes of behaviour in response to changes in the ambient conditions
can be described by models involving switching. In most cases, this switching is assumed
to depend on the system state, and thus it disregards the history and, therefore, memory.
Memory can be introduced into a mathematical model using a phenomenon known as hysteresis.
We illustrate this idea using a simple SIR compartmental model that is applicable in
epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and
how it may be applied to describe a variety of memory effects. Our other objective is to
introduce a unified paradigm for mathematical modelling with memory effects in
epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow
related to large ensembles of elementary exchange operations that recently has been
successfully applied in a number of other areas, such as terrestrial hydrology, and
macroeconomics. For the purposes of illustrating these ideas in an application to biology,
we consider a rather simple case study and develop a model from first principles. We
accompany the model with extensive numerical simulations which exhibit interesting
qualitative effects.